Hurwitz-stable matrix

This article is about matrices whose eigenvalues have negative real part. For the Hurwitz matrices used to check stability of polynomials, see Routh–Hurwitz matrix.

In mathematics, a Hurwitz-stable matrix,^[1] or more commonly simply Hurwitz matrix,^[2] is a square matrix whose eigenvalues all have strictly negative real part. Some authors also use the term stability matrix.^[2] Such matrices play an important role in control theory.

Definition

A square matrix ${\displaystyle A}$ is called a Hurwitz matrix if every eigenvalue of ${\displaystyle A}$ has strictly negative real part, that is,

${\displaystyle \operatorname {Re} [\lambda _{i}]<0\,}$

for each eigenvalue ${\displaystyle \lambda _{i}}$. ${\displaystyle A}$ is also called a stable matrix, because then the differential equation

${\displaystyle {\dot {x}}=Ax}$

is asymptotically stable, that is, ${\displaystyle x(t)\to 0}$ as ${\displaystyle t\to \infty .}$

If ${\displaystyle G(s)}$ is a (matrix-valued) transfer function, then ${\displaystyle G}$ is called Hurwitz if the poles of all elements of ${\displaystyle G}$ have negative real part. Note that it is not necessary that ${\displaystyle G(s),}$ for a specific argument ${\displaystyle s,}$ be a Hurwitz matrix — it need not even be square. The connection is that if ${\displaystyle A}$ is a Hurwitz matrix, then the dynamical system

${\displaystyle {\dot {x}}(t)=Ax(t)+Bu(t)}$

${\displaystyle y(t)=Cx(t)+Du(t)\,}$

has a Hurwitz transfer function.

Any hyperbolic fixed point (or equilibrium point) of a continuous dynamical system is locally asymptotically stable if and only if the Jacobian of the dynamical system is Hurwitz stable at the fixed point.

The Hurwitz stability matrix is a crucial part of control theory. A system is stable if its control matrix is a Hurwitz matrix. The negative real components of the eigenvalues of the matrix represent negative feedback. Similarly, a system is inherently unstable if any of the eigenvalues have positive real components, representing positive feedback.

See also

• M-matrix
• Perron–Frobenius theorem, which shows that any Hurwitz matrix must have at least one negative entry
• Z-matrix